Since approximately the middle of the nineteenth century, there has been a lively interest in geometries based on postulates other than those of Euclid. Mathematicians such as Lobachevski and Riemann investigated the possibility of relaxing Euclid's postulates. One postulate in particular was Euclid's "Parallel Postulate" which states that "For any given point not on a given line there is exactly one line through the point that does not meet the given line."
Relaxing the parallel postulate required showing that any number of lines through a point outside a given line are non-intersecting with the given line, or that none of the lines are intersecting. Such relaxation would result in hyperbolic geometry or curved space. The consequences of such shifts in Euclid's "Parallel Postulate" proved to be spaces of dramatically different structures. For example, in Lobachevskian geometry, which is a hyperbolic geometry, one sees that it is not possible to change the size of a given figure without at the same time changing its shape which implies in effect the existence of an absolute standard of length.
At the same time, a consistent mathematics of geometries of dimensionality greater than three was developed, and there has been during the past century extensive speculation concerning the "fourth dimension." In addition to metric geometries, non-metric geometries have been fully developed, which do not presuppose the concept of length, with the broadest theory of the structure of space, in general, having been formulated in the domain of topology. Differential geometries have made possible theories of curved spaces.
In one application of four-dimensional geometry, the "fourth" dimension is interpreted as time. Einstein's special theory of relativity has made it a commonplace assumption that the fourth dimension is time, while the curvature of physical space or the space-time continuum has become a common concept through popularization of Einstein's general theory of relativity. However, this does not concern geometry per se, which would include a fourth spatial coordinate for describing space.
It was pointed out by Henri Poincare that there is a close association between our intuitive geometrical conceptions, and the behavior of the rigid bodies with which we are familiar. In particular, the human eye may be described as a rigid body moving with the motions of a rigid body in a three-dimensional Euclidean space. It is generally assumed by mathematicians and others that our spatial intuitions are thus shaped and limited, whether by development, inheritance, or our human nature, to those of a 3-D space, and while it is widely granted that we can comprehend figures and relationships in alternative geometries formally and intellectually, it is generally supposed that we cannot visualize them directly. At the same time, it is widely recognized that many mathematicians, as well as students, teachers, engineers and other professional users of applied mathematics, are greatly aided in their understanding of mathematics by the power of the "spatial intuition," whenever this is possible.
"Spatial intuition" is the conventional power of the human visual perception system which, when presented with two dimensional information that carries with it an implication of a third dimension, systematically and as a matter of course, infers the existence of the third dimension. We thus use our insights in one dimension to understand the next or those of higher dimensionality. For example, the retina of our eye has only two dimensions, so whenever we see a three dimensional object, it is because a sequence of two dimensional images has been formed on the retina, from which we intuit the extension in depth of the three dimensional object. We instinctively use such techniques as motions of the head, and such determinations as the orientation of the object to its background to decide where each of the two dimensional images on our retina actually lies with respect to the visual axis, i.e., back, forward, side, top, etc.
What has been perceived to date as the absence of this power in the case of alternative geometries, such as 4-D Euclidean, is often regarded as an important impediment, and many methods have been devised to supplant the missing direct intuition. Such methods include the display of objects in alternative geometries, such as 4-D Euclidean, through the use of projections, intersections or models in conventional space or 3-D Euclidean geometry in a great variety of forms. In general, these take the form of some method of transformation which maps the figure which has been formally defined in an alternative geometry into a figure within 3-D Euclidean space, which we are then able to contemplate in the usual way. In its simplest form, such a figure may consist of an intersection of the figure in the alternative geometry with Euclidean three-dimensional space. Further, however, such figures may take the form of solid, three-dimensional models which can in turn be rotated as solid figures in 3-D space and/or viewed in three-dimensional perspective. The form and complexity of the original figure in the alternative geometry may, by such methods, be revealed through first giving it rotation in its original space, changing the mode or center of projection, in that space and then observing the consequence as shown in the resulting object. Over many years, diagrams or models of this sort have indeed aided greatly in the discussions of alternative geometries.
With the advent of the electronic computer, the power to visualize alternative geometries has been greatly enhanced, as images can now be quickly produced and transformed on demand, and rendered in color under various assumed modes of lighting, and in animation. However, it is important to make the distinction that these images are still only the projections, intersections, or models of objects in 3-D Euclidean space and not the object as it would appear in the alternative geometry itself. Such projections, intersections, and models of objects in 4-D Euclidean geometry have been displayed on computer systems. See, for example, Beyond The Third Dimension: Geometry, Computer Graphics, and Higher Dimension (1990), by Thomas Banchoff, and his film The Hypercube: Projections and Slicing (1978). In these works, Mr. Banchoff defines a 4-D object such as the Hypercube in 4-D space, and then computes and displays corresponding mappings in conventional space either as intersections, orthogonal projections, or stereographic projections. Orthogonal projections are those in which the conventional coordinates are projected unaltered, while the fourth is ignored (corresponding to projection by rays from a source at infinite distance.) Stereographic projections are those in which the three-dimensional object is generated by rays or lines originating from a point at finite distance. Images so generated have been used to produce animated films of the 4-D cube or hypercube, and the 4-D sphere or hypersphere. In these films, as the objects are rotated, dramatic animated sequences are produced in full color and with computer-aided enhancements of light and shading. The resulting images aid greatly in approximating an intuitive sense of the four-dimensional objects themselves. However, these images do not reproduce the effect of light rays coming directly from the object in the alternative geometry, and hence, do not provide a view of the 4-D object itself, but show only the 3-D projections of the 4-D object.
Alternative geometries have a wide range of applications. Within mathematics, for example, four dimensions are required whenever there is a need to map the plane onto itself by means of a topological function, or a number pair into a number pair, as in the graphing of the function of a complex variable. Physicists and engineers frequently work with systems with more variables than three; in general, the graphing of the configuration space of systems of more than three variables generates a locus of higher dimensionality. It is very often valuable to work with curvilinear coordinates, and non-linear systems may often be best expressed in a curvilinear geometry. In order to visualize the curvature of a three-dimensional space properly, however, it is necessary to embed it in a linear space of four dimensions. With the availability of a wealth of computer technology for graphical design and imaging, and even for the experiencing of virtual reality in various modes, it seems clear that an instrument is needed to provide for greater visual intuition of objects in alternative geometries. Such an instrument would need to provide what is in effect a direct visualization of the objects themselves in the alternative geometry, rather than the projections, models and intersections of the prior art.